Given a non-zero quaternion q in H, the map x -> q^-1 * x * q defines an action on the 3-dimensional vector space
of pure quaternions X (ie linear combinations of i,j,k). It turns out that this action is a rotation of X,
and this is a surjective group homomorphism from H* onto SO3. If we restrict q to the group of unit quaternions
(those of norm 1), then this homomorphism is 2-to-1 (since q and -q give the same rotation).
This shows that the multiplicative group of unit quaternions is isomorphic to Spin3, the double cover of SO3.

Given a pair of unit quaternions (l,r), the map x -> l^-1 * x * r defines an action on the 4-dimensional space
of quaternions. It turns out that this action is a rotation, and this is a surjective group homomorphism
onto SO4. The homomorphism is 2-to-1 (since (l,r) and (-l,-r) give the same map).
This shows that the multiplicative group of pairs of unit quaternions (with pointwise multiplication)
is isomorphic to Spin4, the double cover of SO4.